A tree T is said to be homogeneous if it is uniquely rooted and there exists an integer b > 2, called the branching number of T , such that every t ∈ T has exactly b immediate successors. We study the behavior of measurable events in probability spaces indexed by homogeneous trees. Precisely, we show that for every integer b > 2 and every integer n > 1 there exists an integer q(b, n) with the f...

It is shown that every (innnite) graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Moreover, for every nonnegative integer k there is a unique connected graph T(k) that has Cheeger constant k, but removing any edge from it reduces the Cheeger constant. This minimal graph, T(k), is a tree, and every graph G with Cheeger constant h(G) > k has a spanning fore...

Let G = (V, E) be a connected undirected graph, where V = {1, . . . , n} is the set of nodes and E the set of edges. Consider that there are associated positive costs, ce, to each edge e ∈ E. Given a positive integer valued function d : V → N on the nodes, the Min-Degree Constrained Minimum Spanning Tree (md-MST) problem consists in finding a spanning tree T of G with minimum total edge cost, g...

Given a string on an integer alphabet, we present an algorithm that computes the set of all distinct squares belonging to this string in time linear to the string length. As an application, we show how to compute the tree topology of the minimal augmented suffix tree in linear time. Asides from that, we elaborate an algorithm computing the longest previous table in a succinct representation usi...

Abstract Tree-based models are increasingly popular due to their ability identify complex relationships that beyond the scope of parametric models. Survival tree methods adapt these allow for analysis censored outcomes, which often appear in medical data. We present a new Optimal Trees algorithm leverages mixed-integer optimization (MIO) and local search techniques generate globally optimized s...

Let G = (V, E, w) be an undirected graph with nonnegative edge weight. For any spanning tree T of G, the weight of T is the total weight of its tree edges and the routing cost of T is ∑ u,v∈V dT (u, v), where dT (u, v) is the distance between u and v on T . In this paper, we present an algorithm providing a trade off among tree weight, routing cost and time complexity. For any real number α > 1...

A tree cover of a graph G is defined as a collection of trees such that their union includes all the vertices of G. The cost of a tree cover is the weight of the maximum weight tree in the tree cover. Given a positive integer k, the k-tree cover problem is to compute a minimum cost tree cover which has no more than k trees. Star covers are defined analogously. Additionally, we may also be provi...

A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures ...

In this paper the branching trees for attacking MILP are reviewed. Under certain circumstances branches can be done concurrently. This is fully investigated with the result that there are restrictions for certain dual values and reduced costs. As a sideeffect of this study a new class of cuts for MILP is found, which are defined by those values. 1 Motivation of the following thoughts Nowadays t...

A vertex colouring of a graph G is nonrepetitive if for any path P = (v1, v2, . . . , v2r) in G, the first half is coloured differently from the second half. The Thue choice number of G is the least integer l such that for every l-list assignment L of G, there exists a nonrepetitive L-colouring of G. We prove that for any positive integer l, there is a tree T with πch(T ) > l. On the other hand...